Finite Group Databasebeta

← #13144464

Subgroups — #13144464

Order 1728.

715 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 715 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#121C211yesmaximal, characteristic, complemented
2C2×#88545C211yesmaximal, characteristic, complemented
3#11721830C211yesmaximal, characteristic, complemented
4C2×#88538C211yesmaximal, characteristic, complemented
5#11720578C211yesmaximal, characteristic, complemented
6#11720531C211yesmaximal, characteristic, complemented
7#11720530C211yesmaximal, characteristic, complemented
8C3×#5039—34nomaximal
9C2×D4×H(3,𝔽3)—21no—
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C2×#6235—21no—
13C2×C4⋊#272C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C22⋊C4×H(3,𝔽3)—21no—
16#88545—22no—
17#88542—21no—
18C4⋊#1084C2211yescharacteristic
19#88538—22no—
20C2×#6180C2211yescharacteristic
21C2×#6180C2211yescharacteristic
22C32×#121D314yescomplemented
23C2×C3×#213—34no—
24C3×#930—34no—
25C2×C3×#208—34no—
26C3×#832—34no—
27C3×#809—34no—
28C3×#808—34no—
29C23×H(3,𝔽3)—21no—
30C23×H(3,𝔽3)D411yescharacteristic
31C22×#272—21no—
32D4×H(3,𝔽3)—21no—
33D4×H(3,𝔽3)—21no—
34D4×H(3,𝔽3)—42no—
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)—21no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)D411yescharacteristic
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40#6235—42no—
41C2×#1084—21no—
42C4⋊#272—21no—
43C4⋊#272—21no—
44#6180—21no—
45#6180D412yescomplemented
46C3×#469—91no—
47#5039—34no—
48C2×C32×D4—24no—
49C2×C32×D4C2×D314yes—
50C22×C32×C4C2×D314yes—
51C2×C3×#67—64no—
52C2×C3×D12—34no—
53C32×C4⋊C4C2×D314yes—
54C32×C22⋊C4—24no—
55C3×#213—68no—
56C3×(C2×D3)⋊C4—64no—
57C3×C4⋊Dic3—34no—
58C3×#208—68no—
59C2×C3×C3⋊C8—34no—
60C2×C3×C3⋊C8—34no—
61C22×H(3,𝔽3)—21no—
62C22×H(3,𝔽3)—21no—
63C22×H(3,𝔽3)—21no—
64C22×H(3,𝔽3)—41no—
65C22×H(3,𝔽3)—41no—
66C22×H(3,𝔽3)—41no—
67C22×H(3,𝔽3)C2×D411yescharacteristic
68C2×#272—41no—
69C2×#272—41no—
70C2×#272—41no—
71C4×H(3,𝔽3)—21no—
72C4×H(3,𝔽3)—41no—
73C4×H(3,𝔽3)C2×D411yescharacteristic
74C4×H(3,𝔽3)#4111yescharacteristic
75#1084—41no—
76C2×C3×D8—91no—
77C2×C3×M16—91no—
78C3×#121—34no—
79C3×#121—91no—
80C3×#121C3⋊D311yescharacteristic
81C2×#213—34no—
82#930—34no—
83C2×#208—34no—
84C3×#106—91no—
85C3×#102—91no—
86C3×#102—91no—
87#832—34no—
88#809—34no—
89#808—34no—
90C23×C32—24no—
91C23×C32#6714yes—
92C22×C3×D3—64no—
93C32×D4—24no—
94C32×D4—24no—
95C32×D4—48no—
96C2×C32×C4—24no—
97C2×C32×C4—24no—
98C2×C32×C4—24no—
99C2×C32×C4#6714yes—
100C2×C32×C4C22×D314yes—
101C3×#67—128no—
102C2×C3×Dic3—64no—
103C3×D12—64no—
104C3×D12—64no—
105C3×C3⋊C8—38no—
106C3×C3⋊C8—64no—
107#469—91noSylow
108C2×H(3,𝔽3)—41no—
109C2×H(3,𝔽3)—81no—
110C2×H(3,𝔽3)#12111yescharacteristic
111C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
112C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
113#272—81no—
114C2×C3×D4—21no—
115C2×C3×D4—34no—
116C2×C3×D4—64no—
117C2×C3×D4—91no—
118C2×C3×D4—181no—
119C2×C3×D4C2×C3⋊D311yescharacteristic
120C22×C3×C4—34no—
121C22×C3×C4C2×C3⋊D311yescharacteristic
122C2×#67—64no—
123C2×D12—34no—
124C3×D8—182no—
125C3×M16—182no—
126C2×C3×C8—91no—
127C2×C3×C8—91no—
128C3×C4⋊C4—34no—
129C3×C4⋊C4—91no—
130C3×C4⋊C4C2×C3⋊D311yescharacteristic
131C3×C22⋊C4—21no—
132C3×C22⋊C4—64no—
133C3×C22⋊C4—181no—
134#213—68no—
135(C2×D3)⋊C4—64no—
136C4⋊Dic3—34no—
137#208—68no—
138C2×C3⋊C8—34no—
139C2×C3⋊C8—34no—
140C22×C32—24no—
141C22×C32—24no—
142C22×C32—24no—
143C22×C32—44no—
144C22×C32—44no—
145C22×C32—44no—
146C22×C32C2×#6714yes—
147C2×C3×D3—124no—
148C2×C3×D3—124no—
149C2×C3×D3—124no—
150C32×C4—24no—
151C32×C4—44no—
152C32×C4D3×D414yes—
153C32×C4#23714yes—
154C3×Dic3—124no—
155C2×D8—91no—
156C2×M16—91no—
157#121—91no—
158#121#27211yescharacteristic, complemented
159#106—91no—
160#102—91no—
161#102—91no—
162H(3,𝔽3)#46911yescharacteristic, complemented, Sylow
163C23×C3—21no—
164C23×C3—34no—
165C23×C3—64no—
166C23×C3—181no—
167C23×C3#63811yescharacteristic
168C22×D3—64no—
169C3×D4—21no—
170C3×D4—21no—
171C3×D4—42no—
172C3×D4—64no—
173C3×D4—64no—
174C3×D4—128no—
175C3×D4—181no—
176C3×D4—181no—
177C3×D4—362no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—34no—
182C2×C3×C4—34no—
183C2×C3×C4—64no—
184C2×C3×C4—64no—
185C2×C3×C4—64no—
186C2×C3×C4—181no—
187C2×C3×C4#63811yescharacteristic
188C2×C3×C4C22×C3⋊D311yescharacteristic
189#67—128no—
190C2×Dic3—64no—
191D12—64no—
192D12—64no—
193C3×C8—92no—
194C3×C8—181no—
195C3⋊C8—38no—
196C3⋊C8—64no—
197C2×C32—44no—
198C2×C32—84no—
199C2×C32#93214yes—
200C2×C32#93914yes—
201C2×C32#94914yes—
202C3×D3—244no—
203C2×D4—21no—
204C2×D4—91no—
205C2×D4—181no—
206C2×D4C2×#27211yescharacteristic
207C22×C4C2×#27211yescharacteristic
208D8—182no—
209M16—182no—
210C2×C8—91no—
211C2×C8—91no—
212C4⋊C4—91no—
213C4⋊C4C2×#27211yescharacteristic
214C22⋊C4—21no—
215C22⋊C4—181no—
216C22×C3—21no—
217C22×C3—21no—
218C22×C3—21no—
219C22×C3—34no—
220C22×C3—41no—
221C22×C3—41no—
222C22×C3—41no—
223C22×C3—64no—
224C22×C3—64no—
225C22×C3—64no—
226C22×C3—124no—
227C22×C3—124no—
228C22×C3—124no—
229C22×C3—361no—
230C22×C3—361no—
231C22×C3—361no—
232C22×C3C2×#63811yescharacteristic
233C2×D3—124no—
234C2×D3—124no—
235C2×D3—124no—
236C3×C4—21no—
237C3×C4—34no—
238C3×C4—34no—
239C3×C4—41no—
240C3×C4—64no—
241C3×C4—124no—
242C3×C4—361no—
243C3×C4D4×C3⋊D311yescharacteristic
244C3×C4#383011yescharacteristic
245Dic3—124no—
246C32#503914yescomplemented
247C23—21no—
248C23—181no—
249C23#623511yescharacteristic
250D4—21no—
251D4—21no—
252D4—42no—
253D4—181no—
254D4—181no—
255D4—362no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—181no—
260C2×C4#623511yescharacteristic
261C2×C4C22×#27211yescharacteristic
262C8—92no—
263C8—181no—
264C2×C3—34no—
265C2×C3—34no—
266C2×C3—34no—
267C2×C3—41no—
268C2×C3—81no—
269C2×C3—124no—
270C2×C3—244no—
271C2×C3—721no—
272C2×C3#6414111yescharacteristic
273C2×C3#6414811yescharacteristic
274C2×C3#6415811yescharacteristic
275D3—244no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—361no—
283C22—361no—
284C22—361no—
285C22C2×#623511yescharacteristic
286C4—21no—
287C4—41no—
288C4—361no—
289C4D4×#27211yescharacteristic
290C4#8874411yescharacteristic
291C3—34no—
292C3#1059273811yescharacteristic
293C2—41no—
294C2—81no—
295C2—721no—
296C2#1172184211yescharacteristic
297C2#1172184911yescharacteristic
298C2#1172187011yescharacteristic
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#121C2
hypercenterC3×#121C3⋊D3
hypocenterH(3,𝔽3)#469
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#121

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. H(3,𝔽3)×#121C2semidirect
2. C2×#88545C2semidirect
3. #11721830C2semidirect
4. C2×#88538C2semidirect
5. #11720578C2semidirect
6. #11720531C2semidirect
7. #11720530C2semidirect
22. C32×#121D3semidirect
45. #6180D4semidirect
158. #121#272semidirect
162. H(3,𝔽3)#469semidirect
246. C32#5039semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
H(3,𝔽3)×#1212
C2×#885452
#117218302
C2×#885382
#117205782
#117205312
#117205302
C3×#50393